# What is the coefficient of $x^{i}$ in the product $\ \large \prod_{i \geq 1} \frac{1}{1-x^i}\prod_{i \geq 1} \frac{1}{1+x^{2i-1}}$?

What is the coefficient of $$x^{i}$$ in the product $$\ \large \prod_{i \geq 1} \frac{1}{1-x^i}\prod_{i \geq 1} \frac{1}{1+x^{2i-1}}$$?

$$\ \large \prod_{i \geq 1} \frac{1}{1-x^i}\prod_{i \geq 1} \frac{1}{1+x^{2i-1}}$$

=$$\left\{(1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1} \cdots \right\} \left\{(1+x)^{-1}(1+x^3)^{-1} (1+x^5)^{-1} \cdots \right\}$$

=$$(1-x)^{-1}(1+x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1+x^3)^{-1} \cdots$$

But I am at lost right here.

Help me to find the coefficient of $$x^i$$, the general coefficient.

According to Ramanujan (1913) a(n) is close to $$\, (\cosh(x)-\sinh(x)/x)/(4n)$$ where $$\,x:=\pi\sqrt{n}.\,$$
This is only anapproximation whose relative error goes to zero. If you want exact values, there are recursions such as $$a(n) = -2\sum_{m=1}^{\sqrt{n}} (-1)^m a(n-m^2).$$